Equation 677 Database

Linear magma over the extension field F_16 = F_2[α]/⟨α⁴ + α + 1⟩. Operation: x ◇ y = a·x + b·y in F_16 with (a, b) = (1 + α³, α³). Since α + β = (1 + α³) + α³ = 1 in F_16 (char 2), this is the "Type 1" translation-invariant fully-idempotent linear 677 magma, equivalent to x ◇ y = x + β·(y - x) with β = α³. Note β = α³ is a primitive 5th root of unity in F_16* (which has order 15 = 3·5). In characteristic 2 the primitive 10th-root condition collapses to primitive 5th-root since Φ_10(x) = Φ_5(x) mod 2. F_16 is the proper degree-4 extension of F_2; do NOT confuse with the ring Z/16Z (which has zero divisors and is NOT a field). Size 16, fully idempotent, right-cancellative. [text written by Claude]